From 8817ed4b4ca667f77ca2b45b9320fa98a58022fd Mon Sep 17 00:00:00 2001
From: yeye <j.e.garay.labra@rug.nl>
Date: Fri, 9 Jul 2021 15:22:23 +0200
Subject: [PATCH] new figures: paper

---
 kalman/graphics/figure_func.py     |  37 +++++
 kalman/graphics/figure_func2.py    | 170 +++++++++++++++++++++++
 kalman/input_files/aorta.yaml      |  50 +++----
 kalman/input_files/aorta_old.yaml  | 209 -----------------------------
 kalman/input_files/aorta_old2.yaml | 209 -----------------------------
 5 files changed, 233 insertions(+), 442 deletions(-)
 create mode 100644 kalman/graphics/figure_func.py
 create mode 100644 kalman/graphics/figure_func2.py
 delete mode 100755 kalman/input_files/aorta_old.yaml
 delete mode 100755 kalman/input_files/aorta_old2.yaml

diff --git a/kalman/graphics/figure_func.py b/kalman/graphics/figure_func.py
new file mode 100644
index 0000000..d6f06de
--- /dev/null
+++ b/kalman/graphics/figure_func.py
@@ -0,0 +1,37 @@
+import matplotlib.pyplot as plt
+import numpy as np
+from itertools import cycle
+import argparse
+import pickle
+import yaml
+
+
+from matplotlib import rc
+rc('font',**{'family':'sans-serif','sans-serif':['Helvetica']})
+rc('text', usetex=True)
+
+
+u = np.linspace(-2,2,100)
+utrue = 1
+venc1 = 0.9*utrue
+venc2 = 0.6*utrue
+
+fig1, ax1 = plt.subplots(1,1,figsize=(8, 5))
+
+J1 = 1 - np.cos((utrue-u)/venc1*np.pi)
+J2 = 1 - np.cos((utrue-u)/venc2*np.pi)
+
+
+lwidth = 2
+font_size = 28
+
+ax1.plot(u, J1, color = 'orangered', label = '$venc = 0.9 u_{true}$', linestyle='-',linewidth=lwidth)
+ax1.plot(u, J2, color = 'dodgerblue', label = '$venc = 0.6 u_{true}$', linestyle='-',linewidth=lwidth)
+ax1.axvline(x=1,color = 'black',linewidth = lwidth , label = '$u_{true}$')
+
+ax1.legend(fontsize=20, loc= 'upper right')
+ax1.tick_params(axis='both', which='major', labelsize=22)
+ax1.set_yticks([])
+ax1.set_xlabel('$u$',fontsize=font_size)
+plt.show()
+fig1.savefig('functionals.png', dpi=500, bbox_inches='tight')
\ No newline at end of file
diff --git a/kalman/graphics/figure_func2.py b/kalman/graphics/figure_func2.py
new file mode 100644
index 0000000..9afcd10
--- /dev/null
+++ b/kalman/graphics/figure_func2.py
@@ -0,0 +1,170 @@
+import matplotlib.pyplot as plt
+import numpy as np
+from itertools import cycle
+import argparse
+import pickle
+import yaml
+
+
+from matplotlib import rc
+rc('font',**{'family':'sans-serif','sans-serif':['Helvetica']})
+rc('text', usetex=True)
+
+
+fig1, ax1 = plt.subplots(1,1,figsize=(8, 5))
+lwidth = 2
+font_size = 28
+
+################ Flow Parameters
+Rd      =   2.5
+Rt      =   0.5
+GradP   =   4
+mu      =   0.5
+fac     =   1
+nr      =   50
+VENC    =   0.6
+
+gamma   =   267.513e6     #  rad/Tesla/sec   Gyromagnetic ratio for H nuclei
+Bo      =   1.5              #  Tesla           Magnetic Field Strenght
+TE      =   5e-3             #  Echo-time
+M       =   np.ones(nr)         #  Magnetization
+phi0    =   gamma*Bo*TE       # Reference phase
+phi02   =   phi0%3.14
+M1      =   np.pi/(gamma*VENC)
+ff      =   np.pi/(1000*gamma*M1)
+uv      =   np.arange(-4*VENC,4*VENC,ff)
+
+
+r       =   np.linspace(-Rd, Rd, nr)
+dr      =   r[2]-r[1]
+vmax    =   1
+v       =   vmax/Rt**2*( Rt**2 - r**2 )*(np.abs(r)<Rt);  # Poiseuille Formula
+ai      =   v/vmax
+
+theta       =   np.linspace(-4,5,2000)
+vtest       =   np.linspace(-5,5,2000)
+JF          =   0*theta
+jv          =   0*theta
+JV          =   0*theta
+Mjv         =   np.zeros([len(theta),len(ai)])
+jv0         =   0*theta
+JV0         =   0*theta
+Mjv0        =   np.zeros([len(theta),len(ai)])
+#################################### MAGNETIZACION FROM V
+phiv    =   phi02 + v*np.pi/VENC
+modv    =   np.ones(phiv.shape) 
+M1  =   modv*np.cos(phi02)  + 1j*modv*np.sin(phi02)
+M2  =   modv*np.cos(phiv)   + 1j*modv*np.sin(phiv)
+
+################################### FFT to COMPLEX M
+S1      =   np.fft.fft(M1)
+S2      =   np.fft.fft(M2)
+################################### SubSampling
+a1      =   0
+a2      =   1
+##### FILLED WITH ZEROS
+US1         =   S1
+US2         =   0*S2
+US2[a1::a2] =   S2[a1::a2]
+
+
+MR1         =   np.fft.ifft(US1)
+MR2         =   np.fft.ifft(US2)
+vrec1       =   (np.angle(MR2)-phi02)*VENC/(np.pi)
+
+
+for k in range(len(ai)):
+    jv0         =   1-np.cos(np.pi*(vrec1[k]-vtest)/VENC)
+    Mjv0[:,k]   =   jv0[:]
+    JV0         =   JV0 + jv0  
+
+for k in range(len(ai)):
+    jv          =   1-np.cos(np.pi*(vrec1[k]-theta*ai[k])/VENC)
+    Mjv[:,k]    =   jv[:]
+    JV          =   JV + jv 
+
+NJV1    =   JV*100/np.max(JV)
+
+MV = Mjv0
+V =NJV1
+
+left, bottom, width, height = [0.2, 0.2, 0.1, 0.1]
+
+fig     =   plt.figure(figsize=(12, 6), dpi=100)
+ax1     =   plt.subplot(1,2,1)
+
+ch1     =   20
+ch2     =   23
+
+ax0 = fig.add_axes([left, bottom, width, height])
+ax0.plot(r,v,'b-')
+ax0.plot([r[ch1]],[v[ch1]],color='xkcd:coral',marker='o')
+ax0.plot([r[ch2]],[v[ch2]],color='xkcd:azure',marker='o')
+ax0.set_xlim((-1.5,1.5))
+
+
+
+
+#for k in range(22,39):
+#   if k!=ch1 and k!=ch2 and np.sum(MV[:,k])!=0:        
+#       ax1.plot(vtest, MV[:,k],color='xkcd:beige',alpha=0.8)
+
+
+
+ax1.plot(vtest, MV[:,ch1],color='xkcd:coral',label='$v_1$')
+ax1.plot(vtest, MV[:,ch2],color='xkcd:azure',label='$v_2$')
+
+
+m1x     =   vtest[np.where( np.abs(MV[:,ch1] - np.min(MV[:,ch1]))<0.001  )]
+
+m1y     =   np.min(MV[:,ch1])
+
+m2x     =   vtest[np.where( np.abs(MV[:,ch2] - np.min(MV[:,ch2]))<0.001  )]
+#m2x    =   vtest[np.where(MV[:,ch2]==np.min(MV[:,ch2]))]
+m2y     =   np.min(MV[:,ch2])
+
+
+ax1.plot([m1x],[m1y],color='xkcd:coral',marker='o')
+ax1.plot([m2x],[m2y],color='xkcd:azure',marker='o')
+
+
+ax1.axvline(x=v[ch1], color='xkcd:coral', linestyle='--',label='$v_{1,true}$')
+ax1.axvline(x=v[ch2], color='xkcd:azure', linestyle='--',label='$v_{2,true}$')
+
+
+ax1.set_xlabel(r'$u$',fontsize=20)
+ax1.set_ylabel(r'$J_i(u)$',fontsize=20)
+#ax1.legend(loc='upper right', bbox_to_anchor=(0.5, 1.05),ncol=2, fancybox=True, shadow=True,fontsize=15)
+ax1.set_yticks([])
+ax1.set_xticks([])
+ax1.set_xlim((-3.5,3.5))
+ax1.set_ylim((-1.0,2.4))
+
+ax2     =   plt.subplot(1,2,2)
+ax2.plot(theta,V,'b-')
+ax2.axvline(x=1, color='k', linestyle='--')
+ax2.set_xlabel(r'$\theta$',fontsize=20)
+ax2.set_ylabel(r'$J_T(\theta)$',fontsize=20)
+plt.yticks([])
+ax2.set_xticks([])
+plt.title(r'$\theta_{true}=1$' + '\n' +'$venc < v_{max}$',fontsize=15)
+plt.xlim((-2,3))
+
+plt.show()
+
+
+
+
+
+
+#ax1.plot(u, J1, color = 'orangered', label = '$venc = 0.9 u_{true}$', linestyle='-',linewidth=lwidth)
+#ax1.plot(u, J2, color = 'dodgerblue', label = '$venc = 0.6 u_{true}$', linestyle='-',linewidth=lwidth)
+#ax1.axvline(x=1,color = 'black',linewidth = lwidth , label = '$u_{true}$')
+
+
+ax1.legend(fontsize=20, loc= 'upper right')
+ax1.tick_params(axis='both', which='major', labelsize=22)
+ax1.set_yticks([])
+ax1.set_xlabel('$u$',fontsize=font_size)
+plt.show()
+#fig1.savefig('functionals.png', dpi=500, bbox_inches='tight')
\ No newline at end of file
diff --git a/kalman/input_files/aorta.yaml b/kalman/input_files/aorta.yaml
index 0f1c034..aeefea3 100755
--- a/kalman/input_files/aorta.yaml
+++ b/kalman/input_files/aorta.yaml
@@ -5,10 +5,11 @@ fluid:
     density: 1.2
     dynamic_viscosity: 0.035
     stokes: False
-    implicit_windkessel: False
+    implicit_windkessel: True
+    state_velocity: 'update'
 
 io:
-    write_path: 'results/expl_1ms'
+    write_path: 'results/updRz_Pb_wp0.5wm16.84'
     restart:
         path: ''  # './projects/nse_coa3d/results/test_restart2/' 
         time: 0
@@ -27,8 +28,8 @@ boundary_conditions:
         value: ['0','0','-U*sin(DOLFIN_PI*t/Th)*(t<=Th) + (Th<t)*(U*DOLFIN_PI/Th*(t-Th)*exp(-(t-Th)*beta))']
         parameters:
             #U: 75  #REFERENCE
-            U: 120  
-            #U: 150 #Pa/Pb
+            #U: 120  
+            U: 150 #Pa/Pb
             #U: 40 #Pc
             Th: 0.36
             beta: 70
@@ -56,9 +57,9 @@ boundary_conditions:
             #C: 0.0010   # Pb
             #C: 0.0001   # Pc
             #C: 0.0008   # Pg
-            R_d: 7200  # REFERENCE
+            #R_d: 7200  # REFERENCE
             #R_d:  8760  #Pa
-            #R_d: 17520 #Pb x2
+            R_d: 17520 #Pb x2
             #R_d: 4000  #Pc
             p0: 85
             conv: 1333.223874   
@@ -72,9 +73,9 @@ boundary_conditions:
             #C: 0.0010 # Pb
             #C: 0.0001 # Pc
             #C: 0.0008 # Pg
-            R_d: 11520 # REFERENCE
+            #R_d: 11520 # REFERENCE
             #R_d:  8760  #Pa
-            #R_d: 17520 #Pb x2
+            R_d: 17520 #Pb x2
             #R_d: 4000 #Pc
             p0: 85
             conv: 1333.223874   
@@ -87,9 +88,9 @@ boundary_conditions:
             #C: 0.0005 #Pa
             #C: 0.0010 #Pb
             #C: 0.0001 #Pc
-            R_d: 11520 # REFERENCE
+            #R_d: 11520 # REFERENCE
             #R_d:  8760 #Pa
-            #R_d: 17520 #Pb x2
+            R_d: 17520 #Pb x2
             #R_d: 4000 #Pc
             p0: 85
             conv: 1333.223874   
@@ -118,7 +119,7 @@ timemarching:
         flux_report_normalize_boundary: 1
 
     T: 0.8       # end time
-    dt: 0.002
+    dt: 0.001
     write_dt: 0.03
     checkpoint_dt:  0.03 # <= 0: only last; else value + last
     report: 1   # 0: print nothing, 1: print time step and writeout, 2: 1 + flux
@@ -132,7 +133,7 @@ fem:
     convection_skew_symmetric: True    # aka Temam term
     stabilization:
         forced_normal:
-            enabled: False
+            enabled: True
             boundaries: [3,4,5,6]
             gamma: 20
         backflow_boundaries: [3,4,5,6]
@@ -159,27 +160,28 @@ estimation:
         #- 
         #    id: 3
         #    type: 'windkessel'
+        #    mode: 'Rd'
         #    initial_stddev: 1
         - 
             id: 4
             type: 'windkessel'
-            mode: 'RdC'
-            initial_stddev: 1
+            mode: 'Rd'
+            initial_stddev: 0.5
         - 
             id: 5
             type: 'windkessel'
-            mode: 'RdC'
-            initial_stddev: 1
+            mode: 'Rd'
+            initial_stddev: 0.5
         - 
             id: 6
             type: 'windkessel'
-            mode: 'RdC'
-            initial_stddev: 1
+            mode: 'Rd'
+            initial_stddev: 0.5
         - 
             id: 2
             type: 'dirichlet'
             parameters: 'U'
-            initial_stddev: 1
+            initial_stddev: 0.5
 
 
     measurements:
@@ -187,13 +189,13 @@ estimation:
             mesh: '/home/yeye/NuMRI/kalman/meshes/coaortaH3_leo2.0.h5'
             #mesh: './meshes/coaortaH1.h5'
             fe_degree: 1
-            #xdmf_file: 'measurements/aorta_dt1ms/Perturbation/Mg15V70/u_all.xdmf'
-            #file_root: 'measurements/aorta_dt1ms/Perturbation/Mg15V70/u{i}.h5'
-            xdmf_file: 'measurements/aorta_exp_dt1ms/u_all.xdmf'
-            file_root: 'measurements/aorta_exp_dt1ms/u{i}.h5'
+            xdmf_file: 'measurements/aorta_zdir/Perturbation/Mg15V120/u_all.xdmf'
+            file_root: 'measurements/aorta_zdir/Perturbation/Mg15V120/u{i}.h5'
+            #xdmf_file: 'measurements/aorta_exp_dt1ms/u_all.xdmf'
+            #file_root: 'measurements/aorta_exp_dt1ms/u{i}.h5'
             indices: 0 # indices of checkpoints to be processed. 0 == all
             velocity_direction: [0,0,1]
-            noise_stddev: 45     # standard deviation of Gaussian noise
+            noise_stddev: 16.84     # standard deviation of Gaussian noise
     
     roukf:
         particles: 'simplex' # unique or simplex
diff --git a/kalman/input_files/aorta_old.yaml b/kalman/input_files/aorta_old.yaml
deleted file mode 100755
index e9e032b..0000000
--- a/kalman/input_files/aorta_old.yaml
+++ /dev/null
@@ -1,209 +0,0 @@
-
-mesh: './meshes/coaortaH1.h5'
-# Physical parameters of the fluid  
-fluid:
-    density: 1.2
-    dynamic_viscosity: 0.035
-    stokes: False
-    implicit_windkessel: False
-
-io:
-    write_path: 'results/exp_0.1ms'
-    restart:
-        path: ''  # './projects/nse_coa3d/results/test_restart2/' 
-        time: 0
-    write_xdmf: True
-    write_checkpoints: True
-    write_hdf5_timeseries: False
-    write_velocity: 'update'     # update or tentative
-
-boundary_conditions:
-    - 
-        id: 2
-        type: 'dirichlet'
-        #value: ['0','0','-U*sin(DOLFIN_PI*t/Th)*(t<=Th) + (t<0.8)*(Th<t)*(U*DOLFIN_PI/Th*(t-Th)*exp(-(t-Th)*beta)) +
-        #                 -U*sin(DOLFIN_PI*(t-0.8)/Th)*(t<= 0.8+Th )*(t>0.8) + (t<1.6)*(0.8+Th<t)*(U*DOLFIN_PI/Th*(t-0.8-Th)*exp(-(t-0.8-Th)*beta)) +
-        #                 -U*sin(DOLFIN_PI*(t-1.6)/Th)*(t<= 1.6+Th )*(t>1.6) + (t<2.4)*(1.6+Th<t)*(U*DOLFIN_PI/Th*(t-1.6-Th)*exp(-(t-1.6-Th)*beta))' ]
-        value: ['0','0','-U*sin(DOLFIN_PI*t/Th)*(t<=Th) + (Th<t)*(U*DOLFIN_PI/Th*(t-Th)*exp(-(t-Th)*beta))']
-        parameters:
-            #U: 75  #REFERENCE 120
-            U: 100 
-            #U: 150 #Pa/Pb
-            #U: 40 #Pc
-            Th: 0.36
-            beta: 70
-            t: 0
-    - 
-        id: 1
-        type: 'dirichlet'
-        value: ['0','0','0']
-    - 
-        id: 3
-        type: 'windkessel'
-        parameters:
-            R_p: 200
-            C: 0.0004
-            R_d: 4800
-            p0: 85
-            conv: 1333.223874   
-    - 
-        id: 4
-        type: 'windkessel'
-        parameters:
-            #R_p: 480
-            R_p: 480
-            #C: 0.0004  #  REFERENCE
-            C: 0.001  #  REFERENCE
-            #C: 0.0005   # Pa
-            #C: 0.0010   # Pb
-            #C: 0.0001   # Pc
-            #C: 0.0008   # Pg
-            #R_d: 7200  # REFERENCE
-            R_d: 15000  # REFERENCE
-            #R_d:  8760  #Pa
-            #R_d: 17520 #Pb x2
-            #R_d: 4000  #Pc
-            p0: 85
-            conv: 1333.223874   
-    - 
-        id: 5
-        type: 'windkessel'
-        parameters:
-            R_p: 520
-            C: 0.0003 # REFERENCE
-            #C: 0.0005 # Pa
-            #C: 0.0010 # Pb
-            #C: 0.0001 # Pc
-            #C: 0.0008 # Pg
-            #R_d: 11520 # REFERENCE
-            R_d: 40000 
-            #R_d:  8760  #Pa
-            #R_d: 17520 #Pb x2
-            #R_d: 4000 #Pc
-            p0: 85
-            conv: 1333.223874   
-    - 
-        id: 6
-        type: 'windkessel'
-        parameters:
-            R_p: 520
-            C: 0.0003 # REFERENCE
-            #C: 0.0005 #Pa
-            #C: 0.0010 #Pb
-            #C: 0.0001 #Pc
-            R_d: 11520 # REFERENCE
-            #R_d: 35000
-            #R_d:  8760 #Pa
-            #R_d: 17520 #Pb x2
-            #R_d: 4000 #Pc
-            p0: 85
-            conv: 1333.223874   
-
-timemarching:
-    velocity_pressure_coupling: 'fractionalstep'    # monolithic, fractionalstep
-
-    monolithic:
-        timescheme: 'gmp'   # generalized midpoint, steady FIXME TODO
-        theta: 1            # 1: Euler, 0.5: implicit midpoint rule (one-legged)
-        nonlinear:
-            method: 'constant_extrapolation'   # constant_extrapolation, linear_extrapolation, newton, picard, snes
-            maxit: 20
-            init_steps: 30
-            use_aitken: 1   #  0: False, 1: Picard only, 2: all
-            report: 1       # 0: None, 1: residuals, 2: residuals and energy (inflow/driving/forcing via ESSENTIAL Dbcs!)
-            atol: 1.e-6    # note: dot required!!
-            rtol: 1.e-16
-            stol: 0.0
-
-    fractionalstep:
-        scheme: 'CT'      # CT, IPCS
-        coupled_velocity: False     # False faster, True needed if robin_bc implicit
-        robin_bc_velocity_scheme: 'implicit' # explicit, semi-implicit, implicit
-        transpiration_bc_projection: 'robin'      # robin, dirichlet
-        flux_report_normalize_boundary: 1
-
-    T: 0.8       # end time
-    dt: 0.0001
-    write_dt: 0.001
-    checkpoint_dt:  0.001 # <= 0: only last; else value + last
-    report: 1   # 0: print nothing, 1: print time step and writeout, 2: 1 + flux
-
-# solver setup
-fem:
-    velocity_space: p1  # p1 p1b/p1+ p2
-    pressure_space: p1  # p1 p0/dg0 dg1
-
-    strain_symmetric: False
-    convection_skew_symmetric: True    # aka Temam term
-    stabilization:
-        forced_normal:
-            enabled: False
-            boundaries: [3,4,5,6]
-            gamma: 20
-        backflow_boundaries: [3,4,5,6]
-        streamline_diffusion:
-            enabled: False
-            parameter: 'standard'   # standard, shakib, codina, klr
-            length_scale: 'metric' # average, max, metric
-            parameter_element_constant: True
-            Cinv: ~
-        monolithic:
-            infsup: 'pspg'    # pspg, pressure-stabilization
-            graddiv: False
-            consistent: False
-            pressure_stab_constant: 1.
-
-    fix_pressure: False
-    fix_pressure_point: [0., 0. , 0.]
-
-linear_solver:
-    method: 'lu'
-
-estimation:
-    boundary_conditions:
-        #- 
-        #    id: 3
-        #    type: 'windkessel'
-        #    initial_stddev: 1
-        - 
-            id: 4
-            type: 'windkessel'
-            mode: 'RdC'
-            initial_stddev: 1
-        - 
-            id: 5
-            type: 'windkessel'
-            mode: 'RdC'
-            initial_stddev: 1
-        - 
-            id: 6
-            type: 'windkessel'
-            mode: 'RdC'
-            initial_stddev: 1
-        - 
-            id: 2
-            type: 'dirichlet'
-            parameters: 'U'
-            initial_stddev: 1
-
-
-    measurements:
-        -
-            mesh: '/home/yeye/NuMRI/kalman/meshes/coaortaH3_leo2.0.h5'
-            #mesh: './meshes/coaortaH1.h5'
-            fe_degree: 1
-            #xdmf_file: 'measurements/aorta_dt1ms/Perturbation/Mg15V70/u_all.xdmf'
-            #file_root: 'measurements/aorta_dt1ms/Perturbation/Mg15V70/u{i}.h5'
-            xdmf_file: 'measurements/aorta_exp_dt1ms/u_all.xdmf'
-            file_root: 'measurements/aorta_exp_dt1ms/u{i}.h5'
-            indices: 0 # indices of checkpoints to be processed. 0 == all
-            velocity_direction: [0,0,1]
-            noise_stddev: 45     # standard deviation of Gaussian noise
-    
-    roukf:
-        particles: 'simplex' # unique or simplex
-        observation_operator: 'postprocessing'         #state or postprocessing
-        reparameterize: True
-        ODV_functional:
-            enable: False
-            VENC: 138
\ No newline at end of file
diff --git a/kalman/input_files/aorta_old2.yaml b/kalman/input_files/aorta_old2.yaml
deleted file mode 100755
index 1a0fe8d..0000000
--- a/kalman/input_files/aorta_old2.yaml
+++ /dev/null
@@ -1,209 +0,0 @@
-
-mesh: './meshes/coaortaH1.h5'
-# Physical parameters of the fluid  
-fluid:
-    density: 1.2
-    dynamic_viscosity: 0.035
-    stokes: False
-    implicit_windkessel: True
-
-io:
-    write_path: 'results/impl_0.1ms'
-    restart:
-        path: ''  # './projects/nse_coa3d/results/test_restart2/' 
-        time: 0
-    write_xdmf: True
-    write_checkpoints: True
-    write_hdf5_timeseries: False
-    write_velocity: 'update'     # update or tentative
-
-boundary_conditions:
-    - 
-        id: 2
-        type: 'dirichlet'
-        #value: ['0','0','-U*sin(DOLFIN_PI*t/Th)*(t<=Th) + (t<0.8)*(Th<t)*(U*DOLFIN_PI/Th*(t-Th)*exp(-(t-Th)*beta)) +
-        #                 -U*sin(DOLFIN_PI*(t-0.8)/Th)*(t<= 0.8+Th )*(t>0.8) + (t<1.6)*(0.8+Th<t)*(U*DOLFIN_PI/Th*(t-0.8-Th)*exp(-(t-0.8-Th)*beta)) +
-        #                 -U*sin(DOLFIN_PI*(t-1.6)/Th)*(t<= 1.6+Th )*(t>1.6) + (t<2.4)*(1.6+Th<t)*(U*DOLFIN_PI/Th*(t-1.6-Th)*exp(-(t-1.6-Th)*beta))' ]
-        value: ['0','0','-U*sin(DOLFIN_PI*t/Th)*(t<=Th) + (Th<t)*(U*DOLFIN_PI/Th*(t-Th)*exp(-(t-Th)*beta))']
-        parameters:
-            #U: 75  #REFERENCE 120
-            U: 100
-            #U: 150 #Pa/Pb
-            #U: 40 #Pc
-            Th: 0.36
-            beta: 70
-            t: 0
-    - 
-        id: 1
-        type: 'dirichlet'
-        value: ['0','0','0']
-    - 
-        id: 3
-        type: 'windkessel'
-        parameters:
-            R_p: 200
-            C: 0.0004
-            R_d: 4800
-            p0: 85
-            conv: 1333.223874   
-    - 
-        id: 4
-        type: 'windkessel'
-        parameters:
-            #R_p: 480
-            R_p: 480
-            #C: 0.0004  #  REFERENCE
-            C: 0.001  #  REFERENCE
-            #C: 0.0005   # Pa
-            #C: 0.0010   # Pb
-            #C: 0.0001   # Pc
-            #C: 0.0008   # Pg
-            #R_d: 7200  # REFERENCE
-            R_d: 15000  # REFERENCE
-            #R_d:  8760  #Pa
-            #R_d: 17520 #Pb x2
-            #R_d: 4000  #Pc
-            p0: 85
-            conv: 1333.223874   
-    - 
-        id: 5
-        type: 'windkessel'
-        parameters:
-            R_p: 520
-            C: 0.0003 # REFERENCE
-            #C: 0.0005 # Pa
-            #C: 0.0010 # Pb
-            #C: 0.0001 # Pc
-            #C: 0.0008 # Pg
-            #R_d: 11520 # REFERENCE
-            R_d: 40000 
-            #R_d:  8760  #Pa
-            #R_d: 17520 #Pb x2
-            #R_d: 4000 #Pc
-            p0: 85
-            conv: 1333.223874   
-    - 
-        id: 6
-        type: 'windkessel'
-        parameters:
-            R_p: 520
-            C: 0.0003 # REFERENCE
-            #C: 0.0005 #Pa
-            #C: 0.0010 #Pb
-            #C: 0.0001 #Pc
-            R_d: 11520 # REFERENCE
-            #R_d: 35000
-            #R_d:  8760 #Pa
-            #R_d: 17520 #Pb x2
-            #R_d: 4000 #Pc
-            p0: 85
-            conv: 1333.223874   
-
-timemarching:
-    velocity_pressure_coupling: 'fractionalstep'    # monolithic, fractionalstep
-
-    monolithic:
-        timescheme: 'gmp'   # generalized midpoint, steady FIXME TODO
-        theta: 1            # 1: Euler, 0.5: implicit midpoint rule (one-legged)
-        nonlinear:
-            method: 'constant_extrapolation'   # constant_extrapolation, linear_extrapolation, newton, picard, snes
-            maxit: 20
-            init_steps: 30
-            use_aitken: 1   #  0: False, 1: Picard only, 2: all
-            report: 1       # 0: None, 1: residuals, 2: residuals and energy (inflow/driving/forcing via ESSENTIAL Dbcs!)
-            atol: 1.e-6    # note: dot required!!
-            rtol: 1.e-16
-            stol: 0.0
-
-    fractionalstep:
-        scheme: 'CT'      # CT, IPCS
-        coupled_velocity: False     # False faster, True needed if robin_bc implicit
-        robin_bc_velocity_scheme: 'implicit' # explicit, semi-implicit, implicit
-        transpiration_bc_projection: 'robin'      # robin, dirichlet
-        flux_report_normalize_boundary: 1
-
-    T: 0.8       # end time
-    dt: 0.0001
-    write_dt: 0.001
-    checkpoint_dt:  0.001 # <= 0: only last; else value + last
-    report: 1   # 0: print nothing, 1: print time step and writeout, 2: 1 + flux
-
-# solver setup
-fem:
-    velocity_space: p1  # p1 p1b/p1+ p2
-    pressure_space: p1  # p1 p0/dg0 dg1
-
-    strain_symmetric: False
-    convection_skew_symmetric: True    # aka Temam term
-    stabilization:
-        forced_normal:
-            enabled: False
-            boundaries: [3,4,5,6]
-            gamma: 20
-        backflow_boundaries: [3,4,5,6]
-        streamline_diffusion:
-            enabled: False
-            parameter: 'standard'   # standard, shakib, codina, klr
-            length_scale: 'metric' # average, max, metric
-            parameter_element_constant: True
-            Cinv: ~
-        monolithic:
-            infsup: 'pspg'    # pspg, pressure-stabilization
-            graddiv: False
-            consistent: False
-            pressure_stab_constant: 1.
-
-    fix_pressure: False
-    fix_pressure_point: [0., 0. , 0.]
-
-linear_solver:
-    method: 'lu'
-
-estimation:
-    boundary_conditions:
-        #- 
-        #    id: 3
-        #    type: 'windkessel'
-        #    initial_stddev: 1
-        - 
-            id: 4
-            type: 'windkessel'
-            mode: 'RdC'
-            initial_stddev: 1
-        - 
-            id: 5
-            type: 'windkessel'
-            mode: 'RdC'
-            initial_stddev: 1
-        - 
-            id: 6
-            type: 'windkessel'
-            mode: 'RdC'
-            initial_stddev: 1
-        - 
-            id: 2
-            type: 'dirichlet'
-            parameters: 'U'
-            initial_stddev: 1
-
-
-    measurements:
-        -
-            mesh: '/home/yeye/NuMRI/kalman/meshes/coaortaH3_leo2.0.h5'
-            #mesh: './meshes/coaortaH1.h5'
-            fe_degree: 1
-            #xdmf_file: 'measurements/aorta_dt1ms/Perturbation/Mg15V70/u_all.xdmf'
-            #file_root: 'measurements/aorta_dt1ms/Perturbation/Mg15V70/u{i}.h5'
-            xdmf_file: 'measurements/aorta_exp_dt1ms/u_all.xdmf'
-            file_root: 'measurements/aorta_exp_dt1ms/u{i}.h5'
-            indices: 0 # indices of checkpoints to be processed. 0 == all
-            velocity_direction: [0,0,1]
-            noise_stddev: 45     # standard deviation of Gaussian noise
-    
-    roukf:
-        particles: 'simplex' # unique or simplex
-        observation_operator: 'postprocessing'         #state or postprocessing
-        reparameterize: True
-        ODV_functional:
-            enable: False
-            VENC: 138
\ No newline at end of file